Las Vegas Sun newspaper publisher Hank Greenspun was with Frank Sinatra in his apartment at Caesar's Palace in Las Vegas last week when the call came to inform Ol' Blue Eyes that the president's birthday party wouldn't be the same without him.

Hank Greenspun was with Sinatra in Tokyo. He was with Sinatra in Hong Kong; he was with Sinatra when the singer went to Egypt to perform for Mrs. Anwar Sadat in front of the pyramids and he was with Sinatra for concerts in Israel "many, many times."

During the inaugural festivities, Hank Greenspun stayed at the Madison Hotel with other Sinatra stalwarts and went to the party the singer gave there and to the party Sinatra's lawyer, Mickey Rudin, gave the next night at the Jefferson Hotel.

And today, when Sinatra appears before the Nevada Gaming Control Board for a licensing hearing, Greenspun expects to be called as a character witness to recommend that the state approve an application that

Greenspun, a member of the Nevada power-and-politics establishment, said yesterday he will be appearing as a newspaperman who has covered Sinatra for 30 years. He was planning to bring to the hearings copies of all the front page columns written by Greenspun as he "traveled the world" with his buddy during those decades.

Exhibit A, Greenspun said, was going to be a copy of a Las Vegas Sun story in 1963 that was headlined: "Sinatra Pulls Out of Gaming."

Sinatra wasn't "forced out" (because of alleged organized crime associations) as stories then and now have claimed, Greenspun said. "He quit because he was going with Warner Brothers."

According to Greenspun, Sinatra has supplied the gaming regulators with his own FBI files for the hearing.

"The FBI wouldn't release them, so Frank got them under the Freedom of Information Act and turned them over."

Greenspun said that he didn't see anything unusual about a journalist appearing to defend Sinatra. He paid his own way as he traveled the world with the singer, Greenspun said.

"I own this paper . . . I got more money than Frank Sinatra," he said. "I got more money than he ever dreamed of having. I own cable TV, I own half the gold and silver mines in Nevada."

Rudin is Greenspun's lawyer as well as Sinatra's. Three years ago, Greenspun, Rudin and Sinatra were allied on the same side when Sinatra and Rudin attempted unsuccessfully to take over the Del. E. Webb Real Estate Corp., which owns a number of Nevada hotel casinos.

Claire Schweiker, wife of the secretary of health and human services, calls the Chinese red silk dress she wore to President Reagan's birthday party last Friday her consolation prize.

The silk brocade fabric was brought to Mrs. Schweiker two years ago by her husband, who was then in the Senate.

"Two years ago Dick went to China with a delegation from the Senate, and the wives were invited to go along too, but President Carter felt that for us to go would be an unnecessary frill," Mrs. Schweiker explained. "We lobbied, but to no avail.

"Well, Dick knew how disappointed I was, so he brought me this silk brocade fabric back as a consolation. I had it made into a dress by a seamstress in McLean and wear it whenever red's appropriate," she said.

Actress Zsa Zsa Gabor, who never reveals her age, shares the same birthday as the president.

While celebrating her birthday last weekend at a party at Jimmy's in Beverly Hills, Gabor told guests she was waiting for a very important call. She expected that Ronald Reagan, who has always sent her a birthday card, would call.

A disappointed Gabor never received the call.

While Reagan celebrated at his "surprise" party at the White House, Gabor's friends, who had taken over two tables in the middle of Jimmy's restaurant, ate from tins of caviar, placed around the centerpieces of hearts and flowers.

One of Gabor's friends gave her a black negligee that the group wanted her to model, but ever the tease, she said they would have to wait until later. h

The California "group" -- the Reagans' close circle of rich friends, may be used to traveling in limousines, but not all of them are adept at getting out of their chauffeur-driven cars.

James and Gloria Stewart and Irene Dunne, returning to the Madison Hotel from the president's birthday party about 12:15 Saturday morning, spent some time trying to get out of their limousine.

Stewart came out of the back first, opened the front door and helped out Irene Dunne, who had sat next to the president at his party. After struggling to swing her legs forward, she finally stepped out and stood on the sidewalk as Stewart helped his wife from the depths of the limo.

"Finally I'm out -- I thought I'd be stuck in that cave forever," Mrs. Stewart said as she climbed out of the car.